We present an experimental study of macroscopic and microscopic magnetic anisotropy of a spin tetramer system SeCuO3 using torque magnetometry and ESR spectroscopy. Large rotation of macroscopic magnetic axes with temperature observed from torque magnetometry agrees reasonably well with the rotation of the g tensor above T greater than or similar to 50K. Below 50 K, the g tensor is temperature independent, while macroscopic magnetic axes continue to rotate. Additionally, the susceptibility anisotropy has a temperature dependence which can not be reconciled with the isotropic Heisenberg model of interactions between spins. ESR linewidth analysis shows that anisotropic exchange interaction must be present in SeCuO3. These findings strongly support the presence of anisotropic exchange interactions in the Hamiltonian of the studied system. Below T-N = 8 K, the system enters a long-range antiferromagnetically ordered state with easy axis along the <(1) over bar 01 >* direction. Small but significant rotation of magnetic axes is also observed in the antiferromagnetically ordered state suggesting strong spin-lattice coupling in this system.